As PSBC prepares for a $7-8 billion IPO in late 2016 challenges are manifold, including attracting mainland investors while China is clamping down on money flowing offshore.

Vegetable wholesaler Han Chaofeng pulls up outside his local bank branch, parks his three-wheeler truck and uses his postal savings account to send money back to his family in a rural part of Anhui province, some 170 miles away.

“If I want to transfer money to their accounts, they don’t have to walk miles to a bigger town to withdraw it,” says Han who has been using his account at the Postal Savings Bank of...